1. 2.3: Measures of Central Tendency Chapter 2: Descriptive Statistics Objectives... Determine the mean, median, and mode of a population and of a sample Determine the weighted mean of a data set and the mean of a frequency distribution Describe the shape of a distribution as symmetric, uniform, or skewed and compare the mean and median for each Measure of central tendency A value that represents a typical, or central, entry of a data set. Most common measures of central tendency: 1)__________________ 2) __________________ 3) _________________ 1)Mean (average) – _______ of data entries _____________ by the ____________ of _____________. Population mean: Sample mean: Example: Finding a Sample Mean The prices (in dollars) for a sample of roundtrip flights from Chicago, Illinois to Cancun, Mexico are listed. What is the mean price of the flights? 872 432 397 427 388 782 397 Round-off Rules: mean has one more decimal place than set of data values. Rounding should not be done until final answer of calculation.

2. 2) Median - The value that lies in the ____________ of the data when the data set is _____________. Measures the _____________________________________________ If the data set has an  odd number of entries:  even number of entries: Example 2A: Finding the Median The prices (in dollars) for a sample of roundtrip flights from Chicago, Illinois to Cancun, Mexico are listed. Find the median of the flight prices. 872 432 397 427 388 782 397 Example 2B: Finding the Median The flight priced at $432 is no longer available. What is the median price of the remaining flights? 872 397 427 388 782 397 3) Mode The data entry that occurs with the ______________ ______________. If no entry is repeated: If two entries occur with the same greatest frequency, Example 3A: Finding the Mode The prices (in dollars) for a sample of roundtrip flights from Chicago, Illinois to Cancun, Mexico are listed. Find the mode of the flight prices. 872 432 397 427 388 782 397

3. Example 3B: Finding the Mode At a political debate a sample of audience members was asked to name the political party to which they belong. Their responses are shown in the table. What is the mode of the responses? Political PartyFrequency, f Democrat34 Republican56 Other21 Did not respond9 Mode is the only measure of central tendency that can be used to describe data at the _____________ level of measurement. When working with __________________ data the mode is ____________ used. Comparing the Mean, Median, and Mode All three measures describe a typical entry of a data set. Advantage of using the mean: Disadvantage of using the mean: Example 4: Comparing the Mean, Median, and Mode Find the mean, median, and mode of the sample ages of a class shown. Which measure of central tendency best describes a typical entry of this data set? Are there any outliers? Ages in a class 20 21 22 23 24 65

4. Weighted Mean The mean of a data set whose entries have varying weights. Example 5: Finding a Weighted Mean In this class class your grade is determined by two separate weighted averages: 90% for your trimester marking period grade, and 10% for your final exam. Your marking period grade is determined by: 45% for Tests; 35% for Quizzes, and 20% for Homework. Lets say that you have a 75% in homework, an 86% on quizzes and a 82% on tests. What is your trimester marking period grade? If you get a 100% on the final can you get a B+?

5. Mean of a Frequency Distribution Approximated by where x and f are the midpoints and frequencies of a class, respectively, and n is Finding the Mean of a Frequency Distribution In WordsIn Symbols 1. Find the midpoint of each class 2. Find the sum of the products of the midpoints and the frequencies 3. Find the sum of the frequencies (sample size) 4. Find the mean of the distribution Example: Find the Mean of a Frequency Distribution Use the frequency distribution to approximate the mean number of min. that a sample of Internet subscribers spent online in their most recent session. ClassMidpointFreq. f 7 – 186 19 – 3010 31 – 4213 43 – 548 55 – 665 67 – 786 79 – 902 n =

6. _______________ Distribution A vertical line can be drawn through the middle of a graph of the distribution and the resulting halves are approximately mirror images. The Shape of Distributions _____________ Distribution (rectangular) All entries or classes in the distribution have equal or approximately equal frequencies. Symmetric.

7. The Shape of Distributions Contd. __________ ____________ Distribution (negatively skewed) The “tail” of the graph elongates more to the __________. The mean is to the ____________ of the median. ___________ ____________ Distribution (positively skewed) The “tail” of the graph elongates more to the ___________. The mean is to the ___________ of the median.